Alexandroff extension

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In mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named for the Russian mathematician Pavel Alexandrov.

More precisely, let X be a topological space. Then the Alexandroff extension of X is a certain compact space X* together with an open embedding c : X → X* such that the complement of X in X* consists of a single point, typically denoted ∞. The map c is a Hausdorff compactification if and only if X is a locally compact, noncompact Hausdorff space. For such spaces the Alexandroff extension is called the one-point compactification or Alexandroff compactification. The advantages of the Alexandroff compactification lie in its simple, often geometrically meaningful structure and the fact that it is in a precise sense minimal among all compactifications; the disadvantage lies in the fact that it only gives a Hausdorff compactification on the class of locally compact, noncompact Hausdorff spaces, unlike the Stone–Čech compactification which exists for any Tychonoff space, a much larger class of spaces.

Example: inverse stereographic projection[edit]

A geometrically appealing example of one-point compactification is given by the inverse stereographic projection. Recall that the stereographic projection S gives an explicit homeomorphism from the unit sphere minus the north pole (0,0,1) to the Euclidean plane. The inverse stereographic projection S^{-1}: \mathbb{R}^2 \hookrightarrow S^2 is an open, dense embedding into a compact Hausdorff space obtained by adjoining the additional point \infty = (0,0,1). Under the stereographic projection latitudinal circles z = c get mapped to planar circles r = \sqrt{\frac{1+c}{1-c}}. It follows that the deleted neighborhood basis of (1,0,0) given by the punctured spherical caps c \leq z < 1 corresponds to the complements of closed planar disks r \geq \sqrt{\frac{1+c}{1-c}}. More qualitatively, a neighborhood basis at \infty is furnished by the sets S^{-1}(\mathbb{R}^2 
\setminus K) \cup \{ \infty \} as K ranges through the compact subsets of \mathbb{R}^2. This example already contains the key concepts of the general case.

Motivation[edit]

Let c: X \hookrightarrow Y be an embedding from a topological space X to a compact Hausdorff topological space Y, with dense image and one-point remainder \{ \infty \} = Y \setminus c(X). Then c(X) is open in a compact Hausdorff space so is locally compact Hausdorff, hence its homeomorphic preimage X is also locally compact Hausdorff. Moreover, if X were compact then c(X) would be closed in Y and hence not dense. Thus a space can only admit a one-point compactification if it is locally compact, noncompact and Hausdorff. Moreover, in such a one point compactification the image of a neighborhood basis for x in X gives a neighborhood basis for c(x) in c(X), and—because a subset of a compact Hausdorff space is compact if and only if it is closed—the open neighborhoods of \infty must be all sets obtained by adjoining \infty to the image under c of a subset of X with compact complement.

The Alexandroff extension[edit]

Let X be any topological space, and let \infty be any object which is not already an element of X. Put X^* = X \cup \{\infty \}, and topologize X^* by taking as open sets all the open subsets U of X together with all subsets V which contain \infty and such that X \setminus V is closed and compact, (Kelley 1975, p. 150).

The inclusion map c: X \rightarrow X^* is called the Alexandroff extension of X (Willard, 19A).

The above properties all follow from the above discussion:

  • The map c is continuous and open: it embeds X as an open subset of X^*.
  • The space X^* is compact.
  • The image c(X) is dense in X^*, if X is noncompact.
  • The space X^* is Hausdorff if and only if X is Hausdorff and locally compact.

The one-point compactification[edit]

In particular, the Alexandroff extension c: X \rightarrow X^* is a compactification of X if and only if X is Hausdorff, noncompact and locally compact. In this case it is called the one-point compactification or Alexandroff compactification of X. Recall from the above discussion that any compactification with one point remainder is necessarily (isomorphic to) the Alexandroff compactification.

Let X be any noncompact Tychonoff space. Under the natural partial ordering on the set \mathcal{C}(X) of equivalence classes of compactifications, any minimal element is equivalent to the Alexandroff extension (Engelking, Theorem 3.5.12). It follows that a noncompact Tychonoff space admits a minimal compactification if and only if it is locally compact.

Further examples[edit]

  • The one-point compactification of the set of positive integers is homeomorphic to the space consisting of K = {0} U {1/n | n is a positive integer.} with the order topology.
  • The one-point compactification of n-dimensional Euclidean space Rn is homeomorphic to the n-sphere Sn. As above, the map can be given explicitly as an n-dimensional inverse stereographic projection.
  • Since the closure of a connected subset is connected, the Alexandroff extension of a noncompact connected space is connected. However a one-point compactification may "connect" a disconnected space: for instance the one-point compactification of the disjoint union of \kappa copies of the interval (0,1) is a wedge of \kappa circles.
  • The Alexandroff extension can be viewed as a functor from the category of topological spaces to the category whose objects are continuous maps c: X \rightarrow Y and for which the morphisms from c_1: X_1 \rightarrow Y_1 to c_2: X_2 \rightarrow Y_2 are pairs of continuous maps f_X: X_1 \rightarrow X_2, \ f_Y: 
Y_1 \rightarrow Y_2 such that f_Y \circ c_1 = c_2 \circ f_X. In particular, homeomorphic spaces have isomorphic Alexandroff extensions.

See also[edit]

References[edit]

  • P.S. Alexandroff (1924), "Über die Metrisation der im Kleinen kompakten topologischen Räume", Math. Ann. 92 (3-4): 294–301, doi:10.1007/BF01448011 
  • Ronald Brown (1973), "Sequentially proper maps and a sequential compactification", J. London Math Soc., (2) 7: 515–522